{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"1dcea1f12c9e21de4d5214bf2364a3c79a5546bd61d942e3676f77949a51a6b8","created":"2026-10-03T07:17:57Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"ac71e6f07ab8d7c4cf26723f3dbc8f61964399ed19cb24f1250f22102f55d6dd","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"bio.jamming-rigidity.upper-critical-dimension","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"Theory solved exactly in infinitely many dimensions predicts specific numbers for how contacts and forces behave at the jamming point. Simulations find the same numbers in two and three dimensions, and nobody has shown why.","posed_since":"","precise":"For frictionless soft spheres at the jamming density $\\phi_J$, the infinite-dimensional theory predicts the gap distribution g(h) ~ h^(-gamma) with $\\gamma = 0.41269$ and the small-force distribution P(f) ~ f^theta_e with $\\theta_e = 0.42311$ for extended contacts, and excess coordination $\\Delta Z \\sim (\\phi - \\phi_J)^{(1/2)}$. Simulations agree with these values in $d = 2 \\text{ and } 3$, suggesting an upper critical dimension $d_u = 2$. An answer is a controlled argument (renormalization group or rigorous) fixing d_u and computing any corrections in $d = 2, 3$.","problem_ref":null,"references":"","settled_by":"A field-theoretic or rigorous derivation of $d_{u}$ for jamming, confirmed by high-precision finite-size scaling of $\\gamma$ and $\\theta_{e}$ in $d = 2\\ \\text{and}\\ 3$.","status_note":"Numerics agree with mean-field exponents in $d = 2, 3$, but no theory establishes the upper critical dimension (2026).","title":"Are jamming exponents mean-field already in two and three 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